What computers just cannot do. (Part II)

What computers just cannot do.(Part II)COS 116: 3/5/2008Sanjeev Arora

AdministriviaMidterm - in-class 3/13Review session Tues and Wed during lab slot.2006 midterm linked under extras on web

Recap from last time

Turing-Post computational model:Greatly simplified modelInfinite tape, each cell contains 0/1Program = finite sequence of instructions (only 6 types!)Unlike pseudocode, no conditionals or loops, only GOTOcode(P) = binary representation of program P

Example: doubling program1. PRINT 0 2. GO LEFT 3. GO TO STEP 2 IF 1 SCANNED 4. PRINT 1 5. GO RIGHT 6. GO TO STEP 5 IF 1 SCANNED 7. PRINT 1 8. GO RIGHT 9. GO TO STEP 1 IF 1 SCANNED 10. STOP Program halts on this input data if STOP is executed in a finite number of steps

Some factsFact 1: Every pseudocode program can be written as a T-P program, and vice versa

Fact 2: There is a universal T-P program U11001011011100code(P)VU simulates Ps computation on V

Is there a universal pseudocode program ? How would you write it?What are some examples of universal programs in real life?

Halting Problem

Decide whether P halts on V or not

Cannot be solved! Turing proved that no Turing-Post program can solve Halting Problem for all inputs (code(P), V).11001011011100code(P)V

Makes precise something quite intuitive: Impossible to demonstrate a negative Suppose program P halts on input V. How can we detect this in finite time? Just simulate.Intuitive difficulty: If P does not actually halt, no obvious way to detect this after just a finite amount of time.Turings proof makes this intuition concrete.

Ingredients of the proof..

Fundamental assumption: A mathematical statement is either true or false When somethings not right, its wrong. Bob DylanIngredient 1: Proof by contradiction

Aside: Epimenides Paradox Cretans, always liars!But Epimenides was a Cretan!(can be resolved)

More troubling: This sentence is false.

Ingredient 2:Suppose you are given some T-P program PHow would you turn P into a T-P program that does NOT halt on all inputs that P halts on?

Finally, the proofSuppose program H solves Halting Problem on ALL inputs of the form code(P), V.HConsider program DOn input V, check if it is code of a T-P program.If no, HALT immediately.If yes, use doubling program to create the bit string V, V and simulate H on it.If H says Doesnt Halt, HALT immediately.If H says Halts, go into infinite loopGotcha! Does D halt on the input code(D)?If H halts on every input, so does D

Lessons to take awayComputation is a very simple process ( can arise in unexpected places)

Universal Program

No real boundary between hardware, software, and data No program that decides whether or not mathematical statements are theorems. Many tasks are uncomputable; e.g. If we start Game of life in this configuration, will cell (100, 100) ever have a critter?

Age-old mystery: Self-reproduction.How does the seed encode the whole?

Self-ReproductionFallacious argument for impossibility:Blueprint

M.C. Escher

Print Gallery

Fallacy Resolved: Blueprint can involve some computation; need not be an exact copy! Print this sentence twice, the second time in quotes. Print this sentence twice, the second time in quotes.

High-level description of program that self-reproducesPrint 0Print 1...Print 0. . . . . . . . . . . .. . . . . .. . . . . .}Prints binary code of B}Takes binary string on tape, and in its place prints (in English) the sequence of statements that produce it, followed by the translation of the binary string into English.AB

Self-reproducing programs

Fact: for every program P, there exists a program P that has the exact same functionality except at the end it also prints code(P) on the tapeMyDoomCrash users computer!Reproduce program send to someone else!